Method for compensating film height modulations in spin coating of a resist film layer

ABSTRACT

A method compensates film height modulations in spin coating of a resist film layer. From a desired layout pattern, a substrate topography as a result of lithographically structuring in image fields is determined. A spin coating model is provided to determine a modeled resist film height based on the substrate topography during spin coating of a resist film. A nominal resist film height by using the spin coating model with an unperturbed substrate topography having a flat surface is determined. Next, film height modulations based on a difference are determined for test points and the desired layout pattern is optimized by implementing further structural elements in order to form an optimized mask pattern by minimizing the film height modulations.

TECHNICAL FIELD

This invention relates to a method for compensating film heightmodulations in spin coating of a resist film layer.

BACKGROUND

The manufacturing of integrated circuits aims for continuouslydecreasing feature sizes of the fabricated components. Semiconductormanufacturing includes repeatedly projecting a pattern in a lithographicstep onto a semiconductor wafer and processing the wafer to transfer thepattern into a layer deposited on the wafer surface or into thesubstrate of the wafer. This processing includes depositing a resistfilm layer on the surface of the semiconductor substrate in a spincoating process, projecting a mask with the pattern onto the resist filmlayer and developing or etching the resist film layer to create a resiststructure.

The resist structure is transferred into a layer deposited on the wafersurface or into the substrate in an etching step. Planarization andother intermediate processes may further be necessary to prepare aprojection of a successive mask level. Furthermore, the resist structurecan also be used as a mask during an implantation step. The resist maskdefines regions in which the electrical characteristics of the substrateare altered by implanting ions.

The spin coating process can be subdivided in four different stages.First, resist material is deposited on the wafer. Many different ways ofdeposition exist, for a description see D. E. Bornside, C. W. Macosko,and L. E. Scriven, “On the Modeling of Spin Coating”, J. ImagingTechnology, 13, pages 122-130, 1987. In a second stage, the spin-upstep, the wafer rotates and the entire wafer surface is covered withresist liquid. In the third stage, the spin-off step, excess liquid isremoved from the wafer. The liquid flows radially outward and flies offthe edge of the rotating disk. During this stage steep wave fronts ofthe resist liquid can form but they run radially outward. Behind thesefronts a film of nearly uniform thickness is established if the wafersurface is flat.

It is a characteristic feature of the spin coating process that when theresist film continues thinning the film surface has the tendency tobecome more and more uniform. The fourth stage consists of solidifyingthe resist material. Solvent has evaporated during the preceding stagesand, therefore, the resist has meanwhile become so viscous that the lossrate of resist material due to the radial outflow has already reducedmuch. Thereafter, the mass loss due to ongoing solvent evaporationdominates the further thinning of the resist material.

Finally, if the spin coating time has exceeded a certain limit solventevaporation also ceases. Further prolongation of the spinning time hasno significant influence anymore. The film height approaches a steadystate.

Thickness variations of photo resist films are highly undesirable duringchip manufacturing. The reason being that the sizes of the structuresthat are lithographically to be imaged depend sensitively on the resistthickness. Varying resist thicknesses over the wafer can directly impactthe chip yield. Usually, if the wafer substrate is flat, spin coatingyields very uniform resist film surfaces. However, sometimes the wafersubstrate is not flat but shows a distinctive topography. For instance,the necessity to coat a resist film directly on a given layer structurecan arise due to economical reasons because it is time-consuming andexpensive to planarize (to some extent) a wafer surface with anintermediate coating layer.

Experimentally, for spin coating over topographies, it can often beobserved that although the wafer topographies are periodically repeatedinside the chip areas on the wafer, the resist film's thicknessvariations after spin coating are not. The observed thickness varies notonly as a function of the topography inside a single chip area, but alsodepends strongly on the wafer position of the chip. If the resist filmheight varies from chip to chip it becomes impossible tolithographically form the same structures inside the different chipareas on the wafer. The operating parameters during spin coating e.g.spin speed and initial viscosity have to be chosen such that thisundesired wafer signature is minimized.

Besides the experimental observations as described above, manytheoretical studies exist that try to model the spin coating process.

Historically, theoretical spin coating studies were initiated by thework of A. G. Emslie, F. T. Bonner, and L. G. Peck, “Flow of ViscousLiquid on a Rotating Disk”, J. Appl. Phys., Vol. 29, 5, pages 858-862,1958 (hereinafter, “Emslie, et al.”). In this document, the flowbehavior of the resist on a rotating disk is analyzed and thetime-dependent film height is related to the resist flow beneath thesurface. The velocity field inside the resist film has been derived inthe framework of the lubrication approximation of the Navier-Stokesequations and the resist has been treated as an incompressible Newtonianliquid.

Emslie, et al. theoretically explained the experimental fact that a flatfilm surface results when a resist is spun on a flat rotating substrateand that initially non-uniform film profiles tend to become more andmore uniform under centrifugation. Later on, the work of Emslie, et al.has been extended in many respects.

As one of the extensions, the document of S. Middleman, “The effect ofinduced air-flow on the spin coating of viscous liquids”, J. Appl.Phys., Vol. 62, 6, pages 2530-2532, 1987, (hereinafter, “Middleman”)should be mentioned. Importantly, Middleman incorporated the effect ofshear stress at the resist-air interface on the rate of thinning of thefilm. The shearing stress at the resist-air interface results becausethe rotating disk, e.g., a semiconductor wafer, acts like a “centrifugalpump” or fan.

Due to the disk rotation and the friction between the air and the disksurface, the air above the disk gets a velocity component tangential tothe disk circumference. This tangential velocity induces also a radialvelocity component due to the centrifugal acceleration. The radialoutflow of the air in turn results in a vertical air flow towards thedisk. Of special interest is the fact that significant radially andtangentially directed shearing forces are generated by the air flow.

Middleman employed an existing analytical solution of the velocity fieldof the air given by W. G. Cochran, “The flow due to a rotating disk”,Proc. Cambridge Philos. Soc., 30, pages 365-375, 1934 (hereinafter,“Cochran”), and used this analytical expression for the radiallydirected shear stress to show that the shear stress has a significantinfluence on the rate of film thinning. The radially directed shearstress is given by

$\begin{matrix}{\tau_{rz}^{air} = {\frac{1}{2}\omega^{3/2}\mu_{air}^{1/2}\rho_{air}^{1/2}r}} & (1)\end{matrix}$where r is the radial coordinate on the disk, the component τ_(rz)^(air) stands for the r-component of the force per unit area across aplane surface element normal to the z-direction, ω is the angularvelocity of the rotating disk, and μ_(air) and ρ_(air) denote thedynamic viscosity and density of air, respectively.

The expression by Cochran for the tangentially acting stress reads

$\begin{matrix}{{\tau_{\Theta\; z}^{air} = {- 0}},{616\omega^{3/2}\mu_{air}^{1/2}\rho_{air}^{1/2}r}} & (2)\end{matrix}$where Θ is the azimuthal coordinate on the disk and the stress componentτ_(Θz) ^(air) stands for the Θ-component of the force per unit areaacross a plane surface element normal to the z-direction.

The formulas (1) and (2) for the radial and tangential air shear abovethe wafer are accurate as long as a Reynolds-number criterion is met, inthe formR ²ωρ_(air)/μ_(air)<3×10⁵  (3)

For a 300 mm wafer (radius r=15 cm), a spin speed ω=1300 rpm and thekinematic viscosity of air at normal conditions, the Reynolds number is2.04×10⁵, which is not too much under the above limit.

At higher spin speeds (ω>1900 rpm) turbulent air flow above the wafersets in, which would degrade the spin coating performance. It should benoted, that the radial and tangential components τ_(rz) ^(air) andτ_(Θz) ^(air) can as well be expressed in the lateral Cartesiancoordinate basis that is co-rotating with the wafer,

$\begin{matrix}{{\tau_{xz}^{air} = {{R \cdot x} - {T \cdot y}}}{and}{{\tau_{yz}^{air} = {{R \cdot y} + {T \cdot x}}},{where}}{R = {\frac{1}{2}\omega^{3/2}\mu_{air}^{1/2}\rho_{air}^{1/2}}}{and}{T = {{- 0}\text{,}616\omega^{3/2}\mu_{air}^{1/2}\rho_{air}^{1/2}}}} & (4)\end{matrix}$

In the document of D. Meyerhofer, “Characteristics of resist filmsproduced by spinning”, J. Appl. Phys., Vol. 49, 7, pages 3993-3997, 1978(hereinafter, “Meyerhofer”), a spin coating model is presented includingan evaporation rate of solvent during resist spinning. Meyerhoferproposed a solvent evaporation rate that is proportional to the squareroot of the angular velocity. Using this functional dependence of theevaporation rate on the spinning speed, Meyerhofer calculated modelpredicted time-dependent film heights for various spin speeds andcompared to measured values.

In the documents of P. C. Sukanek, “Spin Coating”, J. ImagingTechnology, Vol. 11, 4, pages 184-190, 1985 and P. C. Sukanek, “A modelfor Spin Coating with Topography”, J. Electrochem. Soc., Vol. 136, 10,pages 3019-3026, 1989 (collectively hereinafter, “Sukanek”),Meyerhofer's evaporation approach is further extended by accounting alsofor the dependence of the evaporation rate on solvent content andgas-phase resistance. Sukanek modeled the evaporation rate e with unitsmass/(time×area) ase=αω^(1/2)(ρ_(s)−ρ_(s) ^(res)),  (5)where α is a constant and ρ_(s) denotes the mass density of the solvent.The parameter ρ_(s) ^(res) stands for the density of residual solventthat is known to remain in the film after spin coating. The residualsolvent density in (5) is to be considered as an empirical quantityaccounting approximately for the gas-phase resistance due to saturationabove the film and the finally reduced solvent mobility inside the film.

Sukanek also included surface tension forces and surface tensiongradients in his treatment. In Sukanek's treatment of spin coating overtopographies both the pressure as well as the solvent content becomefunctions of the lateral coordinates in the wafer plane. The surfacetension coefficient and the viscosity depend on the local solventcontent and are also spatially dependent. Sukanek's approach allowedtaking into account the spatial variations of viscosity and surfacetension coefficient.

Other approaches for spin coating over topographical wafer surfaces havebeen disclosed by P. -Y. Wu and F. -C. Chou, “Complete analyticalsolutions of film planarization during spincoating”, J. Electrochem.Soc., Vol. 146, 10, pages 3819-3826, 1999, F. -C. Chou and P. -Y. Wu,“Effect of air shear on film planarization during spin coating”, J.Electrochem. Soc., Vol. 147, 2, pages 699-705, 2000, S. Kim, J. S. Kim,and F. Ma, “On the flow of a thin liquid film over a rotating disk”, J.Appl. Phys., Vol. 69, 4, pages 2593-2601, 1981, and J. S. Kim, S. Kim,and F. Ma, “Topographic effect of surface roughness on thin-film flow”,J. Appl. Phys., Vol. 73, 1, pages 422-428, 1993.

A conceptional advantage compared to other approaches is Sukanek'streatment of solvent evaporation and of local surface tension andviscosity gradients, which are a consequence of topographically inducedperturbations of the solvent flow during spin coating. During the timeevolution these effects are coupled to the film height evolution.

The above-described methods are, however, to some extent approximationsthat rely on certain assumptions. Further investigations might benecessary to provide more detailed results for studying resist filmvariations of a topography on a wafer.

SUMMARY OF THE INVENTION

In one aspect, the invention provides a method for compensating filmheight modulations in spin coating of a resist film layer. It is afurther aspect of the invention to provide methods for compensating filmheight modulations in spin coating of a resist film layer being capableof determining resist film modulations.

A first embodiment of the present invention provides for a method forcompensating film height modulations in spin coating of a resist filmlayer. A desired layout pattern includes a plurality of structuralfeatures, each having characteristic feature sizes. A substratetopography is determined as a result of lithographically structuring asubstrate with the desired layout pattern in a plurality of imagefields. A spin coating model is applicable to determine a modeled resistfilm height based on the substrate topography during spin coating of aresist film. A nominal resist film height is determined by using thespin coating model with an unperturbed substrate topography having aflat surface. A plurality of test points on the substrate are selected.Film height modulations are determined based on a difference of thenominal resist film height and the modeled resist film height for eachof the test points. The desired layout pattern is optimized byimplementing further structural elements in order to form an optimizedmask pattern by minimizing the film height modulations.

Yet another embodiment is provided by a method for compensating filmheight modulations in spin coating of a resist film layer, includingproviding a desired layout pattern comprising a plurality of structuralfeatures each having characteristic feature sizes. A substratetopography is determined as a result of lithographically structuring asubstrate with the desired layout pattern in plurality of image fields.A spin coating model is applicable to determine a modeled resist filmheight based on the substrate topography during a spin coating step of aresist film on the substrate as a function of respective positions onthe substrate. A modeled resist film height is determined for each ofthe respective positions on the substrate, as well as an average resistfilm height based on the nominal resist film heights. Film heightmodulations are determined based on a difference of the average resistfilm height and the modeled resist film height for each of therespective positions, and the desired layout pattern is optimized byimplementing further structural elements in order to form an optimizedmask pattern by minimizing the film height modulations.

Another embodiment is provided by a method for compensating film heightmodulations in spin coating of a resist film layer. A desired layoutpattern is provided including a plurality of structural features eachhaving characteristic feature sizes. A substrate topography isdetermined as a result of lithographically structuring a substrate withthe desired layout pattern in a plurality of image fields, the imagefields having an active area and a surrounding kerf area. A spin coatingmodel is applicable to determine a modeled resist film height based onthe substrate topography during spin coating of a resist film. A nominalresist film height is determined by using the spin coating model with anunperturbed substrate topography having a flat surface. A plurality oftest points on the substrate are selected, and film height modulationsare determined based on a difference of the nominal resist film heightand the modeled resist film height for each of the test points. Thedesired layout pattern is optimized by implementing further structuralelements in order to form an optimized mask pattern by minimizing thefilm height modulations.

Yet another embodiment is provided by a computer program product forcompensating film height modulations in spin coating of a resist filmlayer including computer readable instructions so as to cause a computerto store a desired layout pattern including a plurality of structuralfeatures each having characteristic feature sizes. A substratetopography is determined as a result of lithographically structuring asubstrate with the desired layout pattern in plurality of image fields.A spin coating model is stored, the spin coating model being applicableto determine a modeled resist film height based on the substratetopography during a spin coating step of a resist film on the substrateas a function of respective positions on the substrate. A modeled resistfilm height is determined for each of the respective positions on thesubstrate. An average resist film height is determined based on thenominal resist film heights, and film height modulations are determinedbased on a difference of the average resist film height and the modeledresist film height for each of the respective positions. The desiredlayout pattern is optimized by implementing further structural elementsin order to form an optimized mask pattern by minimizing the film heightmodulations.

Another embodiment is provided by a storage medium being readable for acomputer and having stored computer readable instructions to perform aprogram on the computer for optimizing a photolithographic mask. Adesired layout pattern is stored including a plurality of structuralfeatures each having characteristic feature sizes. A substratetopography is determined as a result of lithographically structuring asubstrate with the desired layout pattern in plurality of image fields.A spin coating model being applicable to determine a modeled resist filmheight based on the substrate topography during a spin coating step of aresist film, is stored on the substrate as a function of respectivepositions on the substrate. A modeled resist film height is determinedfor each of the respective positions on the substrate. An average resistfilm height is determined based on the nominal resist film heights. Filmheight modulations are determined based on a difference of the averageresist film height and the modeled resist film height for each of therespective positions, and the desired layout pattern is optimized byimplementing further structural elements in order to form an optimizedmask pattern by minimizing the film height modulations.

Yet another embodiment is provided by a system for structuring a surfaceof a substrate including a means for providing a desired layout patternincluding a plurality of structural features each having characteristicfeature sizes. A means for determining a substrate topography isprovided as a result of lithographically structuring a substrate withthe desired layout pattern in plurality of image fields. A means forproviding a spin coating model, which is applicable to determine amodeled resist film height based on the substrate topography during spincoating of a resist film is also provided. A means for determining anominal resist film height by using the spin coating model with anunperturbed substrate topography having a flat surface is provided. Ameans for selecting a plurality of test points on the substrate, a meansfor determining film height modulations based on a difference of thenominal resist film height and the modeled resist film height for eachof the test points, and a means for optimizing the desired layoutpattern by implementing further structural elements in order to form anoptimized mask pattern by minimizing the film height modulations arealso provided. A lithographic mask in accordance with the optimized maskpattern, a means for structuring the substrate using the lithographicmask, and a spin coating device for coating a resist film onto thesubstrate are also provided.

BRIEF DESCRIPTION OF THE DRAWINGS

The above features of the present invention will be more clearlyunderstood from consideration of the following descriptions inconnection with accompanying drawings in which:

FIG. 1 depicts a cross sectional view of a resist film on a substratesurface;

FIG. 2 illustrates a result of the spin coating model according to anembodiment of the invention;

FIG. 3 illustrates a result of the spin coating model according to anembodiment of the invention;

FIG. 4 illustrates topography on a substrate according to an embodimentof the invention;

FIGS. 5A to 5C illustrate a result of the spin coating model accordingto an embodiment of the invention;

FIG. 6 illustrates a result of the spin coating model according to anembodiment of the invention;

FIG. 7 illustrates further topography on a substrate according to anembodiment of the invention;

FIGS. 8A and 8B illustrate a result of the spin coating model accordingto an embodiment of the invention; and

FIG. 9 depicts a top view on a substrate surface according to anembodiment of the invention.

DETAILED DESCRIPTION OF ILLUSTRATIVE EMBODIMENTS

A presently preferred embodiment of the methods and the system accordingto the invention is discussed in detail below. It is appreciated,however, that the present invention provides many applicable inventiveconcepts that can be embodied in a wide variety of specific contexts.The specific embodiments discussed are merely illustrative of specificways to apply the method and the system of the invention, and do notlimit the scope of the invention.

In the following, embodiments are described with respect tolithographically structuring a semiconductor wafer by using a resistfilm layer. The invention, however, might also be useful for fabricationof other components that require structuring a surface, like LCD-panelsor the like.

The present first embodiment of the invention focuses mainly on themodeling of wafer signatures for the finally approached steady state. Inthe steady state the equations describing solvent and film heightmodulations are shown to decouple. The steady state does not depend onviscosity and surface tension gradients and also solvent evaporation hasstopped in the steady state. Nevertheless, the presented equations canalso be used for modeling the time evolution of wafer signatures. Forthe time evolution, the coupling between film height and solventfraction inhomogenities is important.

Furthermore, shearing stress due to the air flow above the wafer will betaken into account. The impact of shearing stress on the wafer signatureof the film height modulations is analyzed.

In the following, the basic equations that govern the time and spaceevolution of the film height and the solvent content are derived. First,the governing equation for the film height will be derived. The secondpart deals with the derivation of an evolution equation for the solventmass fraction.

The two main results of the first and second part of this section arepartial differential equations for the evolution of the film height andthe local solvent mass fraction. Both equations are expressed in aCartesian coordinate frame that is co-rotating with the wafer. Theequations for the film height and the solvent fraction are formulated interms of the velocity field beneath the resist film surface.

In the last part the velocity field of the resist on the rotating waferis derived on the basis of the Navier-Stokes equations for anincompressible Newtonian resist liquid. The derived velocity field makesit possible to give concrete formulas for the evolution of the filmheight and the solvent content whose solution is tackled in thefollowing sections.

Employing the appropriate boundary conditions for the velocity field atthe resist film's surface the influences of solvent evaporation, resistfilm curvature and surface tension gradients on the film height and thelocal solvent content will be incorporated.

With respect to FIG. 1, the geometrical situation that appliesthroughout the description is depicted. FIG. 1 shows a portion of theresist film 10 with rigid bottom 20 and time dependent surface 22. Therigid bottom 20 of the topology on a substrate 15 is described by thefunction B(x,y), while the time-dependent surface 22 of the resist film10 is described by H(x,y,t), with x and y being Cartesian coordinates.

In the following, a general relation between the surface 22 of theresist film 10 and the velocity field inside the resist 10 is derived.This relation is the necessary link for constructing the resist surface22 as soon as a solution of the Navier-Stokes equations for the velocityfield has been obtained.

For an incompressible fluid the continuity equation (mass conservation)reduces to ∇·v=0, where v=(u,v,w)^(T) is the vector of the Cartesianvelocity components. Integrating the continuity equation from the bottom20 to the top 22 of the resist yields

$\begin{matrix}{0 = {{\int_{B{({x,y})}}^{H{({x,y,t})}}{\left( {\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z}} \right)\ {\mathbb{d}z}}} = {{w(H)} - {w(B)} + {\int_{B{({x,y})}}^{H{({x,y,t})}}{\left( {\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y}} \right)\ {{\mathbb{d}z}.}}}}}} & (6)\end{matrix}$

While mass conservation holds inside the resist film 10, evaporationcomes into play at the resist-air interface.

At the resist film's top boundaries denoted H and bottom boundariesdenoted B the velocities (u,v,w) fulfill

$\begin{matrix}{{{w(B)} = {{{u(B)}\frac{\partial B}{\partial x}} + {{v(B)}\frac{\partial B}{\partial v}}}}{and}} & (7) \\{\frac{\mathbb{d}H}{\mathbb{d}t} = {{\frac{\partial H}{\partial t} + {{u(H)}\frac{\partial H}{\partial x}} + {{v(H)}\frac{\partial H}{\partial y}}} = {{w(H)} - {\frac{e}{\rho}.}}}} & (8)\end{matrix}$

Equation (7) is a consequence of the most general boundary condition ata rigid surface, where the velocity field at the bottom must beperpendicular to the normal of the rigid surface 20 of the bottom. Thiscondition for the velocity field is the so-called “no penetration”boundary condition, which suffices to derive the evolution equation forthe film height H(x,y,t). Below, the “no slip” boundary condition willbe employed. The resulting “no slip, no penetration” boundary conditionthen states u=v=w=0 at z=B.

In equation (8), ρ denotes the density of the resist fluid 10. For thecase of an evaporating liquid, where e represents the evaporation ratewith units mass/(time×area), equation (8) expresses the kinematicboundary condition for the velocities at the free surface. On account ofthe Leibniz's rule,

$\begin{matrix}{{\frac{\partial}{\partial x}{\int_{B{({x,y})}}^{A{({x,y})}}{{F\left( {x,y,z} \right)}\ {\mathbb{d}z}}}} = {{\int_{B{({x,y})}}^{A{({x,y})}}{\frac{\partial F}{\partial x}\ {\mathbb{d}z}}} + {{F(A)}\frac{\partial A}{\partial x}} - {{F(B)}\frac{\partial B}{\partial x}}}} & (9)\end{matrix}$the integrals appearing in equation (6) can be reformulated. This yieldsfor the depth-integrated continuity equation (6)

$\begin{matrix}{0 = {{w(H)} - {w(B)} + {\frac{\partial}{\partial x}{\int_{B{({x,y})}}^{H{({x,y,t})}}{u\ {\mathbb{d}z}}}} - {{u(H)}\frac{\partial H}{\partial x}} + {{u(B)}\frac{\partial B}{\partial x}} + {\frac{\partial}{\partial y}{\int_{B{({x,y})}}^{H{({x,y,t})}}{v\ {\mathbb{d}z}}}} - {{v(H)}\frac{\partial H}{\partial y}} + {{v(B)}\frac{\partial B}{\partial y}}}} & (10)\end{matrix}$

Inserting the boundary conditions (7) and (8), this becomes a partialdifferential equation relating the surface 22 height H(x,y,t), thevelocity flow rates Q_(x) and Q_(y) and the evaporation rate e

$\begin{matrix}{{\frac{\partial H}{\partial t} = {{{- \frac{\partial}{\partial x}}Q_{x}} - {\frac{\partial}{\partial y}Q_{y}} - \frac{e}{\rho}}}{with}{Q_{x} = {\int_{B}^{H}{u\mspace{7mu}{\mathbb{d}z}}}}{and}{Q_{y} = {\int_{B}^{H}{v\ {{\mathbb{d}z}.}}}}} & (11)\end{matrix}$

The balance equation for the mass density of the solvent in the resistfilm 10 reads

$\begin{matrix}{{\frac{\partial\rho_{s}}{\partial t} + {\nabla{\cdot j}}} = 0} & (12)\end{matrix}$where j is the solvent flux due to convection and diffusion

$\begin{matrix}{j = {{- \underset{\underset{{diffusive}\mspace{14mu}{flux}}{︸}}{D\;{\Delta\rho}_{s}}} + \underset{\underset{{convective}\mspace{14mu}{flux}}{︸}}{v\;\rho_{s}}}} & (13)\end{matrix}$

Here D is the diffusion coefficient and v denotes the velocity vectorv=(u,v,w)^(T). The resist film has generally much smaller extensionalong the z-direction than along the lateral coordinate axes x and y, asthe substrate surface 20 is much larger compared to the thickness of theresist film 10. Consequently, derivatives in x- and y-direction in thediffusion term are negligible compared to the corresponding onescontaining the derivatives with respect to the z-direction.

As a result, it allows the balance equation for the mass density of thesolvent to be approximated as

$\begin{matrix}{{\frac{\partial\rho_{s}}{\partial t} + \frac{\partial\left( {u\;\rho_{s}} \right)}{\partial x} + \frac{\partial\left( {v\;\rho_{s}} \right)}{\partial y} + \frac{\partial\left( {w\;\rho_{s}} \right)}{\partial z}} = {\frac{\partial}{\partial z}{\left( {D\frac{\partial\rho_{s}}{\partial z}} \right).}}} & (14)\end{matrix}$

Integrating equation (14) over the resist thickness and using Leibniz'srule (9) yields to:

$\begin{matrix}{{{\frac{\partial}{\partial t}{\int_{B}^{H}{\rho_{s}{\mathbb{d}z}}}} - {{\rho_{s}(H)}\frac{\partial H}{\partial t}} + {\frac{\partial}{\partial x}{\int_{B}^{H}{\rho_{s}u{\mathbb{d}z}}}} - {{\rho_{s}(H)}{u(H)}\frac{\partial H}{\partial x}} + {{\rho_{s}(B)}{u(B)}\frac{\partial B}{\partial x}} + {\frac{\partial}{\partial y}{\int_{B}^{H}{\rho_{s}v{\mathbb{d}z}}}} - {{\rho_{s}(H)}{v(H)}\frac{\partial H}{\partial y}} + {{\rho_{s}(B)}{v(B)}\frac{\partial B}{\partial y}} + {{\rho_{s}(H)}{w(H)}} - {{\rho_{s}(B)}{w(B)}}} = {D\left( {\frac{\partial\rho_{s}}{\partial z_{{\text{|}z} = H}} - \frac{\partial\rho_{s}}{\partial z_{{\text{|}z} = B}}} \right)}} & (15)\end{matrix}$

Introducing the solvent mass fraction x_(s)=ρ_(s)/ρ, the assumption of aconstant density ρ and using the fact that the velocity vector at thebottom z=B has no component normal to the substrate 15, this equationcan be rewritten as

$\begin{matrix}{{{\frac{\partial}{\partial t}\left( {\left\langle x_{s} \right\rangle\left( {H - B} \right)} \right)} + {\frac{\partial}{\partial x}{\int_{B}^{H}{x_{s}u{\mathbb{d}z}}}} + {\frac{\partial}{\partial y}{\int_{B}^{H}{x_{s}v{\mathbb{d}z}}}} - {{x_{s}(H)}\left( {\underset{\underset{\equiv {{\mathbb{d}H}/{\mathbb{d}t}}}{︸}}{\frac{\partial H}{\partial t} + {{u(H)}\frac{\partial H}{\partial x}} + {{v(H)}\frac{\partial H}{\partial y}}} - {w(H)}} \right)}} = {D\left( {\frac{\partial x_{s}}{\partial z_{{\text{|}z} = H}} - \frac{\partial x_{s}}{\partial z_{{\text{|}z} = B}}} \right)}} & (16)\end{matrix}$where <x_(s)> denotes the z-averaged mass fraction of the solvent.

Next, the boundary conditions must be considered. At the resist film'sbottom z=B, the solvent mass flux vanishes. Since w(B)=0 (no slip, nopenetration) this condition reduces to

${\rho\; D\frac{\partial x_{s}}{\partial z_{{\text{|}z} = H}}} = 0.$

At the resist film's top z=H, two boundary conditions are to beconsidered. The first is the kinematic boundary condition that hasalready been introduced in equation (8):

$\frac{\mathbb{d}H}{\mathbb{d}t} = {w - {\frac{e}{p}.}}$

The surface height H varies due to the vertical velocity component butcan also recede due to evaporation of solvent. The second boundarycondition concerns the solvent flux at the film surface 22

$\begin{matrix}{{\rho\left( {{D\frac{\partial x_{s}}{\partial z}} - {\left( {w - \frac{\mathbb{d}H}{\mathbb{d}t}} \right)x_{s}}} \right)} = {{{- e}\mspace{14mu}{at}\mspace{14mu} z} = H}} & (17)\end{matrix}$stating that the evaporation flux is given by the diffusive andconvective parts of the outward solvent flux expressed in a referenceframe moving with the surface. Combining this solvent flux boundarycondition with the kinematic boundary condition (8) gives

$\begin{matrix}{{D\frac{\partial x_{s}}{\partial z}} = {{- \frac{e}{\rho}}{\left( {1 - {x_{s}(H)}} \right).}}} & (18)\end{matrix}$

Inserting the above in (16) yields

$\begin{matrix}{{{\frac{\partial}{\partial t}\left( {\left\langle x_{s} \right\rangle\left( {H - B} \right)} \right)} + {\frac{\partial}{\partial x}{\int_{B}^{H}{x_{s}u{\mathbb{d}z}}}} + {\frac{\partial}{\partial y}{\int_{B}^{H}{x_{s}v{\mathbb{d}z}}}}} = {- {\frac{e}{\rho}.}}} & (19)\end{matrix}$

Similar to P. C. Sukanek, Spin Coating, J. Imaging Technology, 11, 4,pages 184-190, 1985, this equation can be approximated by replacing themass fraction x_(s) appearing in the integrals with the z-averaged massfraction <x_(s)>:

∫_(B)^(H)x_(s)u𝕕z ≈ ⟨x_(s)⟩Q_(x) and∫_(B)^(H)x_(s)u𝕕z ≈ ⟨x_(s)⟩Q_(x), with Q_(x) = ∫_(B)^(H)u𝕕z andQ_(y) = ∫_(B)^(H)v𝕕z.

With these approximations and on account of the relation for resist filmheight, see equation (11), equation (19) can be expressed as

$\begin{matrix}{\frac{\partial\left\langle x_{s} \right\rangle}{\partial t} = {{- \frac{1}{H - B}}\left\{ {{Q_{x}\frac{\partial\left\langle x_{s} \right\rangle}{\partial x}} + {Q_{y}\frac{\partial\left\langle x_{s} \right\rangle}{\partial y}} + {\frac{e}{\rho}\left( {1 - \left\langle x_{s} \right\rangle} \right)}} \right\}}} & (20)\end{matrix}$

Having established the governing equations for the resist film height Hand the z-averaged solvent fraction <x_(s)>, the flow rates

$\begin{matrix}{Q_{x} = {{\int_{B}^{H}{u{\mathbb{d}z}\mspace{14mu}{and}\mspace{14mu} Q_{y}}} = {\int_{B}^{H}{v{\mathbb{d}z}}}}} & (21)\end{matrix}$have to be specified that enter the equations (11) and (20).

The velocity field u and v necessary for calculating Q_(x) and Q_(y) canbe obtained in the framework of the Navier-Stokes equations for anincompressible Newtonian fluid. Together with the continuity equationthe Navier-Stokes equations form four equations for the four unknownsgiven by the three velocity components u, v, w and the pressure p.

In the rotating coordinate system on the wafer, three forcecontributions enter the Navier-Stokes equations: the Coriolis forces,centrifugal forces and gravitational forces. For the spin coatingproblem Coriolis and gravitational forces, however, are orders ofmagnitude smaller than the centrifugal forces and the force vector canbe approximated by the centrifugal forces.

For the spin coating problem the resist film thickness is much smallerthan the lateral film extension. Under these circumstances the so-called“thin film” or “lubrication” approximation of the Navier-Stokesequations holds. Then the Navier-Stokes equations for the velocities u,v, w inside a thin film on a rotating wafer and the continuity equationsimplify to:

$\begin{matrix}{{{{- {\rho\omega}^{2}}x} = {{{\mu\frac{\partial^{2}u}{\partial z^{2}}} - \frac{\partial p}{\partial x} - {{\rho\omega}^{2}y}} = {{\mu\frac{\partial^{2}v}{\partial z^{2}}} - \frac{\partial p}{\partial y}}}}{0 = \frac{\partial p}{\partial z}}{0 = {\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + {\frac{\partial w}{\partial z}.}}}} & (22)\end{matrix}$

The first two equations can be interpreted as a force balance betweenthe outward directed centrifugal forces on a fluid element and theviscous and pressure forces acting on the surface 22 of a fluid element.The third equation shows that the pressure does not depend on thevertical z-coordinate.

Additionally, the viscosity μ is modeled to depend on the z-averagedsolvent fraction <x_(s)> only. Provided the appropriate boundaryconditions are known at the substrate z=B(x,y) and at the resist surfacez=H(x,y,t), the z-independence of the pressure and the viscosity allowsthe velocity field easily to be constructed by integrating the first twoequations in (22) twice over the resist depth.

At the substrate 20 the boundary conditions for the velocity field read(no slip, no penetration)u=0=v for z=B(x, y)  (23)expressing that the resist fluid moves with the substrate.

At the resist surface the necessary boundary condition can be expressedin terms of the components of the stress tensor normal and tangential tothe surface, respectively. While the normal stress tensor componentsspecify the pressure, the tangential stresses refer to the viscousshearing forces.

For a Newtonian liquid, the viscous shearing stresses can be calculated.In the approximation given by the lubrication theory, the tangentialstress components for the Newtonian resist fluid reduce to:

$\begin{matrix}{\tau_{xz} = {{\mu\frac{\partial u}{\partial z}\mspace{14mu}{and}\mspace{14mu}\tau_{yz}} = {\mu{\frac{\partial v}{\partial z}.}}}} & (24)\end{matrix}$

Above the resist surface the tangential stress tensor components in airare given by equation (4). The differences across the resist surface 22between the tangential stress tensor components in resist 10 and in air14 must be balanced by the tangential component of the surface tensiongradient ∇γ.

The model assumption of a surface tension coefficient γ that dependsonly on the z-averaged solvent fraction γ=γ(<x_(s)>), leads to thefollowing boundary condition for the tangential stresses at theresist-air interface:

$\begin{matrix}{{{\mu\frac{\partial u}{\partial z}} - {Rx} + {Ty}} = {{{\frac{\partial\gamma}{\partial x}\mspace{14mu}{and}\mspace{14mu}\mu\frac{\partial v}{\partial z}} - {Ry} - {Tx}} = \frac{\partial\gamma}{\partial y}}} & (25)\end{matrix}$for z=H, where the constants R and T specifying the stress tensor in airat the resist surface 22 are defined in (4).

Using the above boundary conditions in equation (23) and (25) togetherwith the first two equations of motion in (22), the followingdescription for the velocity field can be derived:

$\begin{matrix}{{{u\left( {x,y,z} \right)} = {\frac{1}{\mu}\begin{Bmatrix}{\left( {z - B} \right)\left( {{Rx} - {Ty} + \frac{\partial\gamma}{\partial x} +} \right.} \\{\left. {\left( {H - B} \right)\left( {{\omega^{2}\rho\; x} - \frac{\partial p}{\partial x}} \right)} \right) -} \\{\frac{\left( {z - B} \right)^{2}}{2}\left( {{\omega^{2}\rho\; x} - \frac{\partial p}{\partial x}} \right)}\end{Bmatrix}}},{{v\left( {x,y,z} \right)} = {\frac{1}{\mu}\begin{Bmatrix}{\left( {z - B} \right)\left( {{Ry} + {Tx} + \frac{\partial\gamma}{\partial y} +} \right.} \\{\left. {\left( {H - B} \right)\left( {{\omega^{2}\rho\; y} - \frac{\partial p}{\partial y}} \right)} \right) -} \\{\frac{\left( {z - B} \right)^{2}}{2}\left( {{\omega^{2}\rho\; y} - \frac{\partial p}{\partial y}} \right)}\end{Bmatrix}}}} & (26)\end{matrix}$

Equations (26) can be integrated over z giving for the velocity fluxes

$\begin{matrix}{{Q_{x} = {\frac{1}{\mu}\left\{ {{\left( {{Rx} - {Ty} + \frac{\partial\gamma}{\partial x}} \right)\frac{\left( {H - B} \right)^{2}}{2}} + {\left( {{\omega^{2}\rho x} - \frac{\partial p}{\partial x}} \right)\frac{\left( {H - B} \right)^{3}}{3}}} \right\}}},{Q_{y} = {\frac{1}{\mu}{\left\{ {{\left( {{Ry} - {Tx} + \frac{\partial\gamma}{\partial y}} \right)\frac{\left( {H - B} \right)^{2}}{2}} + {\left( {{\omega^{2}\rho y} - \frac{\partial p}{\partial y}} \right)\frac{\left( {H - B} \right)^{3}}{3}}} \right\}.}}}} & (27)\end{matrix}$

Inserting these equations into (11) and (20) and using for theevaporation rate e the expression of equation (5), this yields thegoverning equations for the film height:

$\begin{matrix}{\frac{\partial H}{\partial t} = {{\frac{1}{\mu^{2}}\left\{ {{\frac{\left( {H - B} \right)^{2}}{2}\left( {{R\left\lbrack {{x\frac{\partial\mu}{\partial x}} + {y\frac{\partial\mu}{\partial y}}} \right\rbrack} - {T\left\lbrack {{y\frac{\partial\mu}{\partial x}} - {x\frac{\partial\mu}{\partial y}}} \right\rbrack} + {\frac{\partial\mu}{\partial x}\frac{\partial\gamma}{\partial y}} + {\frac{\partial\mu}{\partial y}\frac{\partial\gamma}{\partial y}}} \right)} + {\frac{\left( {H - B} \right)^{3}}{3}\left( {{\omega^{2}\rho\left( {{x\frac{\partial\mu}{\partial x}} + \frac{\partial\mu}{\partial y}} \right)} - {\frac{\partial\mu}{\partial x}\frac{\partial p}{\partial x}} - {\frac{\partial\mu}{\partial y}\frac{\partial p}{\partial y}}} \right)}} \right\}} - {\frac{1}{\mu}\left\{ {{\left( {H - B} \right)\frac{\partial\left( {H - B} \right)}{\partial x}\left( {{Rx} - {Ty} + \frac{\partial\gamma}{\partial x}} \right)} + {\left( {H - B} \right)^{2}\frac{\partial\left( {H - B} \right)}{\partial x}\left( {{\omega^{2}{\rho x}} - \frac{\partial p}{\partial x}} \right)} + {\left( {H - B} \right)\frac{\partial\left( {H - B} \right)}{\partial y}\left( {{Ry} + {Tx} + \frac{\partial\gamma}{\partial y}} \right)} + {\left( {H - B} \right)^{2}\frac{\partial\left( {H - B} \right)}{\partial y}\left( {{\omega^{2}{\rho y}} - \frac{\partial p}{\partial y}} \right)} + {\frac{\left( {H - B} \right)^{2}}{2}\left( {{2R} + \frac{\partial^{2}\gamma}{\partial x^{2}} + \frac{\partial^{2}\gamma}{\partial y^{2}}} \right)} + {\frac{\left( {H - B} \right)^{3}}{3}\left( {{2\omega^{2}\rho} - \frac{\partial^{2}p}{\partial x^{2}} - \frac{\partial^{2}p}{\partial y^{2}}} \right)}} \right\}} - {a\;{\omega^{1/2}\left( {\left\langle x_{s} \right\rangle - x_{s}^{res}} \right)}}}} & (28)\end{matrix}$and the z-averaged solvent fraction:

$\begin{matrix}{\frac{\partial\left\langle x_{s} \right\rangle}{\partial t} = {{{- \frac{1}{\mu}}\left\{ {{\frac{\partial\left\langle x_{s} \right\rangle}{\partial x}\left\lbrack {{\frac{H - B}{2}\left( {{Rx} - {Ty} + \frac{\partial\gamma}{\partial x}} \right)} + {\frac{\left( {H - B} \right)^{2}}{3}\left( {{\omega^{2}{\rho x}} - \frac{\partial p}{\partial x}} \right)}} \right\rbrack} + {\frac{\partial\left\langle x_{s} \right\rangle}{\partial y}\left\lbrack {{\frac{H - B}{2}\left( {{Ry} + {Tx} + \frac{\partial\gamma}{\partial y}} \right)} + {\frac{\left( {H - B} \right)^{2}}{3}\left( {{\omega^{2}{\rho y}} - \frac{\partial p}{\partial y}} \right)}} \right\rbrack}} \right\}} - {\frac{a\;\omega^{1/2}}{H - B}\left( {1 - \left\langle x_{s} \right\rangle} \right)\left( {\left\langle x_{s} \right\rangle - x_{s}^{res}} \right)}}} & (29)\end{matrix}$

It should be noted that these equations are still to be completed by anexpression for the pressure, which can be obtained by considering theboundary condition for the normal stress tensor components. It isconvenient, however, to leave that expression for the momentundetermined. The necessary expression for the pressure will bediscussed below where an approximate solution of the equations (28) and(29) is described using the methods of perturbation theory.

It should also be noted that the equations (28) and (29) explicitlycontain only surface tension gradients (“Marangoni forces”). It is therelation for the pressure to be derived below that will incorporate alsothe “curvature forces” due to surface tension in the model equations.

Additionally, a solution of equations (28) and (29) requires alsoconstitutive equations to be known for the dependence of the surfacetension coefficient γ and the viscosity μ on the z-averaged solventfractions <x_(s)>. On account of such relations γ=γ(<x_(s)>) andμ=μ(<x_(s)>) the derivatives with respect to x and y are to be expressedas

${\frac{\partial\gamma}{\partial q} = {{\frac{\partial\gamma}{\partial\left\langle x_{s} \right\rangle}\frac{\partial\left\langle x_{s} \right\rangle}{\partial q}\mspace{14mu}{and}{\mspace{11mu}\;}\frac{\partial\mu}{\partial q}} = {\frac{\partial\mu}{\partial\left\langle x_{s} \right\rangle}\frac{\partial\left\langle x_{s} \right\rangle}{\partial q}}}},$where q=x, y. (30)

Thus, provided an expression for the pressure is available and theappropriate initial and boundary conditions are given, knowledge of thedependence of the surface tension coefficient γ and the viscosity μ onthe solvent fraction <x_(s)> closes the set of equations (28) and (29).

Next, a calculation of perturbations will be performed. The substratetopography B(x,y) is to be considered as the perturbing quantity due towhich the resist film height H and the z-averaged solvent fraction<x_(s)> deviate from uniformity.

The first order perturbational treatment linearizes the equations (28)and (29) with respect to the perturbations of the film height and thesolvent fraction. This linearization will be useful for the analysis ofthe steady state to which the film height and solvent fraction tend. Theresult of the steady state analysis will be a “spin coating kernelfunction” that can be used to calculate the film height modulations thatresult after spin coating over topographies.

The substrate topography is written in the formB(x,y)=α·φ(x,y),  (31)where α denotes the perturbation parameter. Generally, the perturbationparameter α is in the range between 0 and 1. The modeled resist filmheight H, the solvent fraction <x_(s)> and the pressure can be expandedasH(x,y,t)=H ₀(t)+α·h ₁(x,y,t)+O(α²),(x _(s))(x,y,t)=x ₀(t)+α·x ₁(x,y,t)+O(α²),p(x,y)=P ₀ +α·P ₁(x,y)+O(α²),where O(α²) denotes terms of order α² and the unperturbed (α=0 ) filmheight H₀ and solvent fraction x₀ have been assumed to be positionindependent. The coordinate independence of the unperturbed pressure P₀then follows as a consequence.

Now the expansions in terms of the perturbation parameter are insertedin the equations (28) and (29), which yields equations for the differentperturbational orders. Here, only the first two orders are considered todescribe the evolution of the modeled resist film height H and thesolvent fraction <x_(s)>, which is a valid approximation for asufficiently small perturbation parameter.

For the unperturbed film height and solvent fraction, the followingresult is determined:

$\begin{matrix}{{\frac{\mathbb{d}H_{0}}{\mathbb{d}t} = {{\frac{- 1}{\mu\left( x_{0} \right)}\left( {{H_{0}^{2}R} + {\frac{2}{3}H_{0}^{3}\omega^{2}\rho}} \right)} - {a\;{\omega^{1/2}\left( {x_{0} - x_{s}^{res}} \right)}}}}{and}{\frac{\mathbb{d}x_{0}}{\mathbb{d}t} = {\frac{{- a}\;\omega^{1/2}}{H_{0}}\left( {1 - x_{0}} \right){\left( {x_{0} - x_{s}^{res}} \right).}}}} & (32)\end{matrix}$

In order to obtain the equations corresponding to order O(α) thefollowing Taylor expansions in terms of α are determined:

${\left. {\frac{\left( {1 - \left\langle x_{s} \right\rangle} \right)\left( {\left\langle x_{s} \right\rangle - x_{s}^{res}} \right)}{H - B} = {{{\frac{1}{H_{0}}\left( {1 - x_{0}} \right)\left( {x_{0} - x_{s}^{res}} \right)} + {\frac{\alpha}{H_{0}}\left\{ {{x_{1}\left( {1 - {2x_{0}} + x_{s}^{res}} \right)} - {\frac{h_{1} - \phi}{H_{0}}\left( {1 - x_{0}} \right)\left( {x_{0} - x_{s}^{res}} \right)}} \right\}} + {{O\left( \alpha^{2} \right)}\frac{1}{\mu\left( \left\langle x_{s} \right\rangle \right)}{f\left( \left\langle x_{s} \right\rangle \right)}}} = {{\frac{1}{\mu\left( x_{0} \right)}{f\left( x_{0} \right)}} + {\alpha\left\{ {\frac{1}{\mu\left( x_{0} \right)}\frac{\partial f}{\partial\alpha}} \right._{\alpha = 0}} - {\frac{x_{1}}{{\mu\left( x_{0} \right)}^{2}}\frac{\partial\mu}{\partial x_{0}}}}}} \right\} + {O\left( \alpha^{2} \right)}},{\frac{\partial\gamma}{\partial q} = {{\alpha\frac{\partial\gamma}{\partial x_{0}}\frac{\partial x_{1}}{\partial q}} + {O\left( \alpha^{2} \right)}}},{\frac{\partial\mu}{\partial q} = {{\alpha\frac{\partial\mu}{\partial x_{0}}\frac{\partial x_{1}}{\partial q}} + {O\left( \alpha^{2} \right)}}},{with}$q = x, y.

Inserting the perturbation expansions in terms of α for the modeledresist film height H and the solvent fraction <x_(s)> into equations(28) and (29) and equating all terms of order O(α) yields for the firstorder perturbations h₁ and x₁ of the film height

$\begin{matrix}{\frac{\partial h_{1}}{\partial t} = {{\frac{H_{0}^{2}}{2\mu^{2}}\frac{\partial\mu}{\partial x_{0}}\left\{ {{R_{1}\left( {{x\frac{\partial x_{1}}{\partial x}} + {y\frac{\partial x_{1}}{\partial y}}} \right)} + {2R_{1}x_{1}} - {T\left( {{y\frac{\partial x_{1}}{\partial x}} - {x\frac{\partial x_{1}}{\partial y}}} \right)}} \right\}} - {\frac{H_{0}}{\mu}\left\{ {{R_{2}\left( {{x\frac{\partial\left( {h_{1} - \phi} \right)}{\partial x}} + {y\frac{\partial\left( {h_{1} - \phi} \right)}{\partial y}}} \right)} + {2{R_{2}\left( {h_{1} - \phi} \right)}} - {T\left( {{y\frac{\partial\left( {h_{1} - \phi} \right)}{\partial x}} - {x\frac{\partial\left( {h_{1} - \phi} \right)}{\partial y}}} \right)} + {\frac{H_{0}}{2}\frac{\partial\gamma}{\partial x_{0}}\left( {\frac{\partial^{2}x_{1}}{\partial x^{2}} + \frac{\partial^{2}x_{1}}{\partial y^{2}}} \right)} - {\frac{H_{0}^{2}}{3}\left( {\frac{\partial^{2}P_{1}}{\partial x^{2}} + \frac{\partial^{2}P_{1}}{\partial y^{2}}} \right)}} \right\}} - {a\;\omega^{1/2}x_{1}}}} & (33)\end{matrix}$and the solvent fraction

$\begin{matrix}{\frac{\partial x_{1}}{\partial t} = {{{- \frac{H_{0}}{2\mu}}\left\{ {{R_{1}\left( {{x\frac{\partial x_{1}}{\partial x}} + {y\frac{\partial x_{1}}{\partial y}}} \right)} - {T\left( {{y\frac{\partial x_{1}}{\partial x}} - {x\frac{\partial x_{1}}{\partial y}}} \right)}} \right\}} + {\frac{a\;\omega^{1/2}}{H_{0}}\left( {{\frac{h_{1} - \phi}{H_{0}}\left( {1 - x_{0}} \right)\left( {x_{0} - x_{s}^{res}} \right)} - {x_{1}\left( {1 - {2x_{0}} + x_{s}^{res}} \right)}} \right)}}} & (34)\end{matrix}$where the abbreviations

$R_{1} = {{\left( {R + {\frac{2}{3}H_{0}\omega^{2}\rho}} \right)\mspace{11mu}{and}{\mspace{14mu}\;}R_{2}} = \left( {R + {H_{0}\omega^{2}\rho}} \right)}$have been introduced.

Next, the expression for the pressure perturbation P₁(x,y) needs to bedetermined. To evaluate the pressure perturbation P₁(x,y), the balancefor the normal stress tensor components at the resist-air interface areemployed. The normal stress tensor boundary condition correct up to O(α)reads:

$\begin{matrix}{{p = {p_{ext} + {2\mu\frac{\partial w}{\partial z}} - {\gamma\left( {\frac{\partial^{2}H}{\partial x^{2}} + \frac{\partial^{2}H}{\partial y^{2}}} \right)}}}{at}{{z = H},}} & (35)\end{matrix}$where p_(ext) stands for the external pressure. On account of thecontinuity equation, using equation (26) for evaluating up to firstorder terms and inserting the result in (35) yields finally for thepressure perturbation P₁(x,y):

$\begin{matrix}{P_{1} = {{- {\gamma\left( {\frac{\partial^{2}h_{1}}{\partial x^{2}} + \frac{\partial^{2}h_{1}}{\partial y^{2}}} \right)}} - {4\left( {R + {H_{0}\omega^{2}\rho}} \right)\left( {h_{1} - \phi} \right)} - {2H_{0}\omega^{2}{\rho\left( {{x\frac{\partial\left( {h_{1} - \phi} \right)}{\partial x}} + {y\frac{\partial\left( {h_{1} - \phi} \right)}{\partial y}}} \right)}} - {2H_{0}\frac{\partial\gamma}{\partial x_{0}}\left( {\frac{\partial^{2}x_{1}}{\partial x^{2}} + \frac{\partial^{2}x_{1}}{\partial y^{2}}} \right)} + {2\frac{H_{0}}{\mu}\frac{\partial\mu}{\partial x_{0}}\left( {R + {\frac{H_{0}}{2}\omega^{2}\rho}} \right)\left( {{x\frac{\partial x_{1}}{\partial x}} + {y\frac{\partial x_{1}}{\partial y}}} \right)} + {2{R\left( {{x\frac{\partial\phi}{\partial x}} + {y\frac{\partial\phi}{\partial y}}} \right)}} + {H_{0}^{2}\left( {\frac{\partial^{2}P_{1}}{\partial x^{2}} + \frac{\partial^{2}P_{1}}{\partial y^{2}}} \right)}}} & (36)\end{matrix}$

This is a partial differential equation for the pressure perturbationP₁(x,y). In order to solve equations (32 to 34) and (36) appropriateinitial and boundary conditions are necessary.

In the following the governing equations (33), (34) and (36) for theperturbations of the film height h₁, the solvent fraction x₁ and thepressure P₁ will be restricted to a small area. For the lubricationapproximation still to hold, the areas must be much larger in extensionthan the resist thickness. On the other hand, the areas are to be sosmall that the differential operators in the governing equations can bereplaced by a center position of the respective small area underconsideration.

This replacement can be justified by introducing the scaled anddimensionless coordinates ξ and η:x=r _(max) ·ξ, y=r _(max) ·η, r _(max)=√{square root over (|x| _(max) ²+|y| _(max) ²)},  (37)wherein r_(max), |x|_(max) ² and |y|_(max) ² denote the maximum radialcoordinate and the maximum absolute values of the lateral x and ycoordinates, respectively, inside the small area under consideration.

The so-defined dimensionless scaled coordinates ξ and η are bothrestricted to the values in the range between −1 and 1. Inside the smallarea, the x and y coordinates can be expressed by the center coordinatesx and y plus a displacement δx and δy, respectively, asx= x+δx and y= y+δy, where |δx|<Δx and |δy|<Δy.

Also the center coordinates x and y and the displacements δx and δy canbe expressed in scaled coordinates asx=r _(max) ξ, y=r _(max) η, δx=r _(max)δξ, and δy=r _(max)δη.

In terms of these scaled coordinates the differential operators inequation (33, 34) and (36) become

${{x\frac{\partial}{\partial x}} + {y\frac{\partial}{\partial y}}} = {\underset{\underset{{\overset{\_}{x}\frac{\partial}{\partial x}} + {\overset{\_}{y}\frac{\partial}{\partial y}}}{︸}}{{\overset{\_}{\xi}\frac{\partial}{\partial\xi}} + {\overset{\_}{\eta}\frac{\partial}{\partial\eta}}} + {{\delta\xi}\frac{\partial}{\partial\xi}} + {{\delta\eta}\frac{\partial}{\partial\eta}}}$and${{y\frac{\partial}{\partial x}} - {x\frac{\partial}{\partial y}}} = {\underset{\underset{{\overset{\_}{y}\frac{\partial}{\partial x}} - {\overset{\_}{x}\frac{\partial}{\partial y}}}{︸}}{{\overset{\_}{\eta}\frac{\partial}{\partial\xi}} - {\overset{\_}{\xi}\frac{\partial}{\partial\eta}}} + {{\delta\xi}\frac{\partial}{\partial\xi}} - {{\delta\xi}{\frac{\partial}{\partial\eta}.}}}$

If the small area under consideration is located far enough from thewafer center, the scaled displacements both are very small compared tounity|δξ|<<1 and |δη|<<1.

Under this condition the replacement

$\begin{matrix}{{{{x\frac{\partial}{\partial x}} + {y\frac{\partial}{\partial y}}} \approx {{\overset{\_}{x}\frac{\partial}{\partial x}} + {\overset{\_}{y}\frac{\partial}{\partial y}}}}{and}{{{y\frac{\partial}{\partial x}} - {x\frac{\partial}{\partial y}}} \approx {{\overset{\_}{y}\frac{\partial}{\partial x}} - {\overset{\_}{x}\frac{\partial}{\partial y}}}}} & (38)\end{matrix}$is a valid approximation.

A typical size of the small areas is approximately 0.05 cm. Thus, forsmall areas of this size whose center coordinates are located fartherthan, say 1 cm, from the wafer 15 center the replacements (38) can beexpected to be a good approximation.

These preliminaries allow a Fourier expansion of the governing equationsinside the small areas. The lateral coordinates inside the small areasare given by the center coordinates x and y and small displacements δxand δy .

The following Fourier expansions are used:

${\phi\begin{pmatrix}{\overset{\_}{x} +} \\{{\delta\; x},{\overset{\_}{y} +}} \\{\delta\; y}\end{pmatrix}} = {\int{\int_{- \infty}^{\infty}{{\mathbb{d}v_{x}}{\mathbb{d}v_{y}}{f\left( {v_{x},v_{y}} \right)}{\exp\left( {2{{\pi\mathbb{i}}\begin{pmatrix}{{v_{x}\left\lbrack {\overset{\_}{x} + {\delta\; x}} \right\rbrack} +} \\{v_{y}\left\lbrack {\overset{\_}{y} + {\delta\; y}} \right\rbrack}\end{pmatrix}}} \right)}}}}$ ${{h_{1}\begin{pmatrix}{\overset{\_}{x} +} \\{{\delta\; x},{\overset{\_}{y} +}} \\{{\delta\; y},t}\end{pmatrix}} = {\int{\int_{- \infty}^{\infty}{{\mathbb{d}v_{x}}{\mathbb{d}v_{y}}{{\overset{\sim}{h}}_{1}\left( {v_{x},v_{y},t} \right)}{\exp\left( {2{{\pi\mathbb{i}}\begin{pmatrix}{{v_{x}\left\lbrack {\overset{\_}{x} + {\delta\; x}} \right\rbrack} +} \\{v_{y}\left\lbrack {\overset{\_}{y} + {\delta\; y}} \right\rbrack}\end{pmatrix}}} \right)}}}}},{{x_{1}\begin{pmatrix}{\overset{\_}{x} +} \\{{\delta\; x},{\overset{\_}{y} +}} \\{{\delta\; y},t}\end{pmatrix}} = {\int{\int_{- \infty}^{\infty}{{\mathbb{d}v_{x}}{\mathbb{d}v_{y}}{{\overset{\sim}{x}}_{1}\left( {v_{x},v_{y},t} \right)}{\exp\left( {2{{\pi\mathbb{i}}\begin{pmatrix}{{v_{x}\left\lbrack {\overset{\_}{x} + {\delta\; x}} \right\rbrack} +} \\{v_{y}\left\lbrack {\overset{\_}{y} + {\delta\; y}} \right\rbrack}\end{pmatrix}}} \right)}}}}},{{P_{1}\begin{pmatrix}{\overset{\_}{x} +} \\{{\delta\; x},{\overset{\_}{y} +}} \\{\delta\; y}\end{pmatrix}} = {\int{\int_{- \infty}^{\infty}{{\mathbb{d}v_{x}}{\mathbb{d}v_{y}}{{\overset{\sim}{p}}_{1}\left( {v_{x},v_{y}} \right)}{\exp\left( {2{{\pi\mathbb{i}}\begin{pmatrix}{{v_{x}\left\lbrack {\overset{\_}{x} + {\delta\; x}} \right\rbrack} +} \\{v_{y}\left\lbrack {\overset{\_}{y} + {\delta\; y}} \right\rbrack}\end{pmatrix}}} \right)}}}}},$where v_(x) and v_(y) are spatial frequencies.

For the small area under consideration only the fixed values x and yenter the equations (33, 34) and (36). Therefore, the Fourier expansionslead to a set of ordinary differential equations for the Fouriercoefficients. In terms of the Fourier coefficients the equation for thepressure perturbation becomes

$\begin{matrix}{{{{\overset{\sim}{p}}_{1}\left( {v_{x},v_{y}} \right)} = \frac{{{\overset{\sim}{x}}_{1} \cdot A} + {{\overset{\sim}{h}}_{1} \cdot B} + {f \cdot C}}{1 + {\left( {2\pi} \right)^{2}{H_{0}^{2}\left( {v_{x}^{2} + v_{y}^{2}} \right)}}}},} & (39)\end{matrix}$with the following abbreviations:

${{A\left( {v_{x},v_{y},\overset{\_}{x},\overset{\_}{y}} \right)} = {H_{0}\left\{ {{\frac{2\pi\; i}{\mu}\frac{\partial\mu}{\partial x_{0}}2{R_{3}\left( {{v_{x}\overset{\_}{x}} + {v_{y}\overset{\_}{y}}} \right)}} + {\left( {2\pi} \right)^{2}\frac{\partial\gamma}{\partial x_{0}}2\left( {v_{x}^{2} + v_{y}^{2}} \right)}} \right\}}},{{B\left( {v_{x},v_{y},\overset{\_}{x},\overset{\_}{y}} \right)} = {{{\gamma\left( {2\pi} \right)}^{2}\left( {v_{x}^{2},v_{y}^{2}} \right)} - {4R_{2}} - {2\pi\;{i \cdot 2}H_{0}\omega^{2}{\rho\left( {{v_{x}\overset{\_}{x}} + {v_{y}\overset{\_}{y}}} \right)}}}},{{C\left( {v_{x},v_{y},\overset{\_}{x},\overset{\_}{y}} \right)} = {{4R_{2}} + {2\pi\;{i \cdot 2}{R_{2}\left( {{v_{x}\overset{\_}{x}} + {v_{y}\overset{\_}{y}}} \right)}}}}$and $R_{3} = {\left( {R + {\frac{1}{2}H_{0}\omega^{2}\rho}} \right).}$

For each pair of spatial frequencies v_(x) and v_(y), the equations forthe Fourier coefficients of the perturbations of the film height and thesolvent fraction can be cast into matrix form:

$\begin{matrix}{{\frac{\mathbb{d}}{\mathbb{d}t}\begin{pmatrix}{{\overset{\sim}{h}}_{1}\left( {v_{x},v_{y},t} \right)} \\{{\overset{\sim}{x}}_{1}\left( {v_{x},v_{y},t} \right)}\end{pmatrix}} = {{\begin{pmatrix}D & E \\F & G\end{pmatrix}\begin{pmatrix}{\overset{\sim}{h}}_{1} \\{\overset{\sim}{x}1}\end{pmatrix}} - {{f\left( {v_{x},v_{y}} \right)}\begin{pmatrix}H \\F\end{pmatrix}}}} & (40)\end{matrix}$where the coefficients are given by:

${D\left( {v_{x},v_{y},\overset{\_}{x},\overset{\_}{y}} \right)} = {{\frac{- H_{0}}{\mu}\left\{ {{R_{2}\left( {2 + {2\pi\;{i\left( {{v_{x}\overset{\_}{x}} + {v_{y}\overset{\_}{y}}} \right)}}} \right)} - {2\pi\;{i \cdot T \cdot \left( {{v_{x}\overset{\_}{y}} - {v_{y}\overset{\_}{x}}} \right)}}} \right\}} - {\frac{H_{0}}{3\mu}\frac{{H_{0}^{2}\left( {2\pi} \right)}^{2}\left( {v_{x}^{2} + v_{y}^{2}} \right)}{1 + {{H_{0}^{2}\left( {2\pi} \right)}^{2}\left( {v_{x}^{2} + v_{y}^{2}} \right)}}{B\left( {v_{x},{v_{y}\overset{\_}{x}},\overset{\_}{y}} \right)}}}$${E\left( {v_{x},v_{y},\overset{\_}{x},\overset{\_}{y}} \right)} = {{{- a}\;\omega^{1/2}} + {\frac{H_{0}^{2}}{2\mu^{2}}\frac{\partial\mu}{\partial x_{0}}\left\{ {{R_{1}\left( {2 + {2\pi\;{i\left( {{v_{x}\overset{\_}{x}} + {v_{y}\overset{\_}{y}}} \right)}}} \right)} - {2\pi\;{i \cdot T \cdot \left( {{v_{x}\overset{\_}{y}} - {v_{y}\overset{\_}{x}}} \right)}}} \right\}} + {\frac{H_{0}^{2}}{2\mu}\frac{\partial\gamma}{\partial x_{0}}\left( {2\pi} \right)^{2}\left( {v_{x}^{2} + v_{y}^{2}} \right)} - {\frac{H_{0}}{3\mu}\frac{{H_{0}^{2}\left( {2\pi} \right)}^{2}\left( {v_{x}^{2} + v_{y}^{2}} \right)}{1 + {{H_{0}^{2}\left( {2\pi} \right)}^{2}\left( {v_{x}^{2} + v_{y}^{2}} \right)}}{A\left( {v_{x},v_{y},\overset{\_}{x},\overset{\_}{y}} \right)}}}$$F = {\frac{a\;\omega^{1/2}}{H_{0}^{2}}\left( {1 - x_{0}} \right)\left( {x_{0} - x_{s}^{res}} \right)}$${G\left( {v_{x},v_{y},\overset{\_}{x},\overset{\_}{y}} \right)} = {{\frac{{- 2}\pi\;{iH}_{0}}{2\mu}\left\{ {{R_{1} \cdot \left( {{v_{x}\overset{\_}{x}} + {v_{y}\overset{\_}{y}}} \right)} - {T \cdot \left( {{v_{x}\overset{\_}{y}} - {v_{y}\overset{\_}{x}}} \right)}} \right\}} - {\frac{a\;\omega^{1/2}}{H_{0}}\left( {1 - {2x_{0}} + x_{s}^{res}} \right)}}$${H\left( {v_{x},v_{y},\overset{\_}{x},\overset{\_}{y}} \right)} = {{\frac{- H_{0}}{\mu}\left\{ {{R_{2}\left( {2 + {2\pi\;{i\left( {{v_{x}\overset{\_}{x}} + {v_{y}\overset{\_}{y}}} \right)}}} \right)} - {2\pi\;{i \cdot T \cdot \left( {{v_{x}\overset{\_}{y}} - {v_{y}\overset{\_}{x}}} \right)}}} \right\}} + {\frac{H_{0}}{3\mu}\frac{{H_{0}^{2}\left( {2\pi} \right)}^{2}\left( {v_{x}^{2} + v_{y}^{2}} \right)}{1 + {H_{0}^{2}\left( {v_{x}^{2} + v_{y}^{2}} \right)}}{C\left( {v_{x},v_{y},\overset{\_}{x},\overset{\_}{y}} \right)}}}$

The coefficients A, B and C are given below the expression (39) for theFourier coefficients of the pressure perturbation.

The ordinary differential equations (40) for the Fourier coefficientscan be solved numerically together with the equations (32) for theunperturbed film height and unperturbed solvent fraction. However, aquantitative modeling of the time evolution requires not only initialconditions to be known but also the functional dependence of the surfacetension coefficient and the viscosity on the solvent fraction.

Now, a solution for the steady state is constructed. Experimentally itis known that the film height and solvent fraction approach finally astate that is practically independent of a further prolongation of thespin coating time. This steady state is approached if the spin coatingperiod lasts long enough. Then, the final film thickness and the finalsolvent fraction will be reached.

In the steady state, equations (40) simplify to

$\begin{matrix}{{{\begin{pmatrix}D & E \\F & G\end{pmatrix}\begin{pmatrix}{\overset{\sim}{h}}_{1} \\{\overset{\sim}{x}}_{1}\end{pmatrix}} = {{f\left( {v_{x},v_{y}} \right)}\begin{pmatrix}H \\F\end{pmatrix}}},} & (41)\end{matrix}$whose solution reads

$\begin{matrix}{\begin{pmatrix}{\overset{\sim}{h}}_{1} \\{\overset{\sim}{x}}_{1}\end{pmatrix} = {\frac{f\left( {v_{x},v_{y}} \right)}{{D \cdot G} - {E \cdot F}}{\begin{pmatrix}{{H \cdot G} - {E \cdot F}} \\{{{- F} \cdot H} + {D \cdot F}}\end{pmatrix}.}}} & (42)\end{matrix}$

Additionally, equation (32) for the unperturbed solvent fraction x₀gives in the steady state

$\begin{matrix}{{\frac{\mathbb{d}x_{0}}{\mathbb{d}t}\overset{!}{=}{0 = {\frac{{- a}\;\omega^{1/2}}{H_{0}}\left( {1 - x_{0}} \right)\left( {x_{0} - x_{s}^{res}} \right)}}},} & (43)\end{matrix}$which shows that the coefficient F approaches zero in the steady state.

Then, it follows also that the Fourier coefficients of the perturbationof the solvent fraction vanish. Thus, the steady state is characterizedby a uniform solvent distribution.

In contrast to the uniform solvent distribution the result for theFourier coefficients of the film height modulation becomes in the steadystate

$\begin{matrix}{{{\overset{\sim}{h}}_{1}\left( {v_{x},v_{y}} \right)} = {\frac{H\left( {v_{x},v_{y}} \right)}{\underset{\underset{\overset{\sim}{S}{({v_{x},v_{y}})}}{︸}}{D\left( {v_{x},v_{y}} \right)}}{f\left( {v_{x},v_{y}} \right)}}} & (44)\end{matrix}${tilde over (S)}(v_(x),v_(y)) describes the frequency response of thefilm height modulation. Using the explicit expressions for H and Dresults in

$\begin{matrix}{{\overset{\sim}{S}\left( {v_{x},v_{y}} \right)} = \frac{{6R_{2}} + {2R_{2}{Q\left( {v_{x}^{2} + v_{y}^{2}} \right)}} + {i\left\lbrack {{av}_{x} + {bv}_{y} + {{Q\left( {v_{x}^{2} + v_{y}^{2}} \right)}\left( {{cv}_{x} + {dv}_{y}} \right)}} \right\rbrack}}{{\Gamma\left( {v_{x}^{2} + v_{y}^{2}} \right)}^{2} + {6R_{2}} + {2R_{2}{Q\left( {v_{x}^{2} + v_{y}^{2}} \right)}} + {i\left\lbrack {{av}_{x} + {bv}_{y} + {{Q\left( {v_{x}^{2} + v_{y}^{2}} \right)}\left( {{ev}_{x} + {gv}_{y}} \right)}} \right\rbrack}}} & (45)\end{matrix}$where the following abbreviations have been usedΓ=γ(2π)⁴ H ₀ ²,Q=H ₀ ²(2π)²,a( x, y )=6π(R ₂ x−T y ),b( x, y )=6π(R ₂ y+T x ),c( x, y )=2π(R ₂ x−3T y ),d( x, y )=2π(R ₂ y+3T x ),e( x, y )=2π((3R ₂−2H ₀ω²ρ) x −3T y ),g( x, y )=2π((3R ₂−2H ₀ω²ρ) y +3T x ), andR ₂ =R+H ₀ω²ρ.

Interestingly, the only material parameters of the resist 10 that enterthe expression for the frequency response are the resist density and thesurface tension coefficient for the finally approached solvent content.In particular, the steady state frequency response is independent of thefinal viscosity. In the steady state, the equation for the film heightmodulation decouples completely from the solvent content and from thegradients of surface tension and viscosity. Only the time evolution isconcerned by these quantities.

The relation given by the frequency response between the Fouriercomponents of the film height modulations and the Fourier components ofthe topographical perturbation corresponds to a convolution in theposition space

$\begin{matrix}{{{H\left( {{\overset{\_}{x} + {\delta\; x}},{\overset{\_}{y} + {\delta\; y}}} \right)} = {H_{0} + \underset{\underset{\alpha\; h_{1{({{\overset{\_}{x} + {\delta\; x}},{\overset{\_}{y} + {\delta\; y}}})}}}{︸}}{\int{\int_{- \infty}^{\infty}{{\mathbb{d}\xi}{\mathbb{d}\eta}\;{{B\left( {{\overset{\_}{x} + {\delta\; x} - \xi},{\overset{\_}{y} + {\delta\; y} - \eta}} \right)} \cdot {S\left( {\xi,\eta,\overset{\_}{x},\overset{\_}{y}} \right)}}}}}}},} & (46)\end{matrix}$where H( x+δx, y+δy) denotes the film height at a position withcoordinates δx and δy relative to the center coordinates x and y of thesmall area under consideration.

The film height is given by the unperturbed film height in the steadystate plus the convolution of the local topography with the local “spincoating kernel function” that depends parametrically on the centercoordinates. The perturbational order parameter α does not appear in thefinal expression for the film height anymore.

The spin coating kernel is given as the inverse Fourier transform of thefrequency response

$\begin{matrix}{{S\left( {{\delta\; x},{\delta\; y},\overset{\_}{x},\overset{\_}{y}} \right)} = {\int{\int_{- \infty}^{\infty}{{\mathbb{d}v_{x}}{\mathbb{d}v_{y}}{\overset{\sim}{S}\left( {v_{x},v_{y},\overset{\_}{x},\overset{\_}{y}} \right)}{{\exp\left( {2{{\pi\mathbb{i}}\left( {{v_{x}\delta\; x} + {v_{y}\delta\; y}} \right)}} \right)}.}}}}} & (47)\end{matrix}$

The spin coating kernel is normalized according to

$\begin{matrix}{{\int{\int_{- \infty}^{\infty}\ {{\mathbb{d}\delta}\; x{\mathbb{d}\delta}\;{{yS}\left( {{\delta\; x},{\delta\; y},\overset{\_}{x},\overset{\_}{y}} \right)}}}} = {{\overset{\sim}{S}\left( {0,0} \right)} = 1.}} & (48)\end{matrix}$

Accordingly, the spin coating kernel function has the meaning of a“point spread function” describing the resulting film height modulationthat is induced by a topographical point defect at x and y.

By numerically (inverse) Fourier transforming the frequency responseaccording to equation (45), the lateral extent of the spin coatingkernel can be determined. Assuming a typical side length of the smallarea under consideration of, say 0.05 cm, this then allows checking theconsistency of the small area approximation for the steady state.

For each center position, the spin coating kernel function ischaracterized by six parameters only. The first two are operatingparameters of the spin coating process, i.e., spin speed and unperturbedfilm height. Secondly, depends on two material properties of the resist10, which are given by the mass density and the finally approachedsurface tension coefficient of the resist 10.

Also, the spin coating kernel depends on the density and the viscosityof the air 14 above the wafer. Except for the final surface tensioncoefficient γ these parameters are well known. The final surface tensioncoefficient γ has to be determined experimentally, either by directmeasurement or by fitting the predicted film height modulations toexperimental data. A typical value for surface tension coefficient γ forphoto resists is 30 dyn/cm, with 1 dyn=10⁻⁵ N.

The following example calculations use the values γ=30 dyn/cm for thefinal surface tension coefficient and ρ=1.017 g/cm³ for the resistdensity. The viscosity μ_(air)=1.81 10⁻⁴ g/(sec×cm) and the density ofair ρ_(air)=1.25×10⁻³ g/cm³ are used. The final unperturbed resistthickness has been chosen to be 440 nm and the spin speed is 1300 rpm.

A typical figure of the modulus of the frequency response correspondingto the above parameters is shown in FIG. 2. The modulus of the frequencyresponse as given in equation (45) is plotted for the center positionx=−5 cm and y=5 cm.

The modulus of the frequency response {tilde over (S)}(v_(x), v_(y)) isshown at including the effect of air shear above the wafer as obtainedfrom equation (45).

The corresponding convolution kernel S(δx, δy, x, y) can be obtained bynumerically Fourier transforming equation (45). FIG. 3 shows the “spincoating convolution kernel” corresponding to FIG. 2.

The lateral extent of the convolution kernel describes its range ofinfluence. For the given example kernel with center coordinates locatedat x=−5 cm and y=5 cm, this range can be estimated to be in the rangebetween 0.005 to 0.01 cm.

Topographical substrate structures that are farther distant than thisrange do not significantly influence the film height modulations at thecenter position.

FIG. 2 and FIG. 3 show that the air shear tilts both, the frequencyresponse as well as the spin coating kernel and the direction ofnarrowest extent is tilted. This effect is due to the tangentiallyacting air drag. Furthermore the extension of the spin coating kernelfunction is reduced. The existence of the tilt and the spatialconcentration of the spin coating kernel function due to the air shearabove the wafer 15 is a general feature that holds at all waferpositions.

The spatial concentration of the spin coating kernel function is anundesired feature, since small spin coating kernels correspond to morepronounced film height modulations. A blurred spin coating kernelfunction will be mostly preferred since blurring of the spin coatingkernel function corresponds to an averaging of the film heightmodulations.

Referring now to FIG. 4, a first topography 26 is considered as anexample. FIG. 4 shows a top view onto the example topography 26 onsubstrate 15 with square blocks 28 of area 30 μm×30 μm and a height of100 nm in z-direction. The resulting film height modulations after spincoating over this topography 26 are predicted for different waferpositions x, y and different spin speeds.

To visualize the position dependency of the film height modulations,three wafer positions are considered along a radius arm. The three waferpositions are i) x=1 cm and y=1 cm, ii) x=5 cm and y=5 cm, and iii) x=10cm and y=10 cm.

Close to the wafer center ( x=1 cm and y=1 cm), see FIG. 5A, the spincoating kernel has a relatively wide spatial extension.

At x=5 cm and y=5 cm, the spin coating kernel becomes narrower, see FIG.5B.

Close to the wafer edge along the radius arm the spin coating kernel hasits narrowest shape, see FIG. 5C.

By comparison of FIGS. 5A to 5C, it can be seen that the air shearinduced tapering of the spin coating kernel along the radius armcorresponds to more and more pronounced film height modulations.

In another embodiment of the invention, the initial solvent content ofthe photo resist 10 is varied. If the initial solvent content isincreased this corresponds to a reduced initial viscosity. Although theactual resist chemistry is rather complex, a simple model for the impactof a reduced initial viscosity at constant (unperturbed) film height H₀would predict that the final (steady state) viscosity remained the samefor the steady state, no matter which initial viscosity had beenprepared. Also, if the initial viscosity is to be changed the spin speedhas to be adjusted in order to obtain the same final (unperturbed)resist height. Reducing the initial viscosity thus necessitates areduction of the spin speed.

Therefore, the effect of reducing the initial viscosity corresponds tosetting the spin speed in formula (45) to a smaller value.

In order to demonstrate the effect, the spin speed is set to 800 rpm.The resulting spin coating kernel close to the wafer center ( x=1 cm andy=1 cm) is shown in FIG. 6. Accordingly, the spin coating kernel has arelatively smaller spatial extension as compared to FIG. 4.

As a result, a reduced initial viscosity corresponds to a blurred spincoating kernel and thus to less pronounced film height modulations.Thus, film height modulations can be minimized by reducing the initialresist viscosity. However, an arbitrary dilution of the resist 10 cannotbe achieved. The spin coating process requires a minimum resistviscosity for obtaining acceptable coating homogenities.

Application of the spin coating kernel of the spin coating model basedon equations (45) and (46) revealed some basic trends. First, filmheight modulations become more pronounced for increasing radialpositions on the wafer. Second, reducing the initial viscosity(corresponding to a reduced spin speed) results in a flattening of thefilm height modulations.

The application of the spin coating kernel to particular topographicallayer structures allows determining the film height modulations thatresult after spin coating of photo resist on such a layer. Thispossibility of application is of particular interest for “implantlayers” where knowledge of the film height at specific sites inside thechip is a prerequisite to calculate the size of the implant structuresthat are lithographically to be formed.

Another determination of modeled resist film heights concerns thedependency of wafer signatures of film heights on the azimuthal waferposition. In particular topographical structures that show a strongasymmetrical orientation, e.g., an isolated line parallel to thecoordinate axis, result in wafer signatures that show not only a radialbut also a distinctively azimuthal position dependency over the wafer.This is a consequence of both, the dependency of the spatial extent ofthe spin coating kernel on the radial wafer coordinate and the deviationof its shape from radial symmetry around the center of the respectivesmall wafer area.

Referring now to FIG. 7, an isolated line structure (width=15 μm,height=100 nm) as an example for a topographical structure 26 is shownresulting in wafer signatures that are strongly dependent on theazimuthal wafer position.

The resulting modeled resist film heights are shown in FIG. 8A and FIG.8B for x=0 cm and y=8 cm and for x=8 cm and y=0 cm, respectively. Themaximum resist film height modulation is in the first case 37 nm and inthe second case 50 nm.

In order to minimize the undesired resist film modulations ontopographical signatures, further structural elements can be included inthe desired layout pattern.

Usually, the substrate topography is lithographically structured inplurality of juxtaposed image fields 30, as shown in FIG. 9. Each imagefield 30 has an active area 40 defining the electrical function of theproduced circuit and a surrounding kerf area 42 which contains alignmentand/or overlay marks and the like.

In a first step, several test points 50 are selected on the substrate.The test points 50 can be selected within the kerf area to define interchip test points. The test points can also be selected within the activearea to define intra chip test points.

Next, film height modulations are determined based on the difference ofthe unperturbed resist film height and the modeled resist film heightfor each of the test points 50.

The desired layout pattern is then optimized by implementing the furtherstructural elements 60 in order to form an optimized mask pattern. Thisis performed by minimizing the film height modulations. The furtherstructural elements 60 can be associated to the kerf area 42, as shownin FIG. 9 and/or to the active area 40.

Then, a lithographic mask can be provided in accordance with theoptimized mask pattern. Consequently, the further structural elementsare identical for each of image field 30.

It is, however, also conceivable to implement partially differentfurther structural elements 60 for different image fields. This wouldrequire to either provide different lithographic masks or to provide anexchangeable sub-mask for the further structural elements 60.

By using the lithographic mask the substrate can be structures. This canbe performed by a photolithographic projection or by using electron beamlithography.

Afterwards, a resist film 10 is spin coated onto the substrate 15, asexplained above. The film height predictions that are made possible bythe derived spin coating kernel function are a necessary prerequisitefor such a “spin coating proximity correction”.

Having described embodiments of the invention, it is noted thatmodifications and variations can be made by persons skilled in the artin light of the above teachings. It is therefore to be understood thatchanges may be made in the particular embodiments of the inventiondisclosed that are within the scope and spirit of the invention asdefined by the appended claims.

Having thus described the invention with the details and theparticularity required by the patent laws, what is claimed and desiredto be protected by Letters Patent is set forth in the appended claims.

1. A method for compensating film height modulations in spin coating ofa resist film layer, the method comprising: providing a desired layoutpattern comprising a plurality of structural features, each structuralfeature having a characteristic feature size; determining a substratetopography as a result of lithographically structuring a substrate withsaid desired layout pattern in a plurality of image fields; providing aspin coating model, said spin coating model being applicable todetermine a modeled resist film height based on said substratetopography during spin coating of a resist film; determining a nominalresist film height by using said spin coating model with an unperturbedsubstrate topography having a flat surface; selecting a plurality oftest points on said substrate; determining film height modulations basedon a difference of said nominal resist film height and said modeledresist film height for each of said test points; and optimizing saiddesired layout pattern by implementing further structural elements inorder to form an optimized mask pattern by minimizing said film heightmodulations.
 2. The method according to claim 1, further comprising:providing a lithographic mask in accordance with said optimized maskpattern; structuring said substrate using said lithographic mask; andspin coating a resist film onto said substrate.
 3. The method accordingto claim 2, wherein prior to providing the spin coating model thefollowing steps are performed: selecting a resist film being used duringthe spin coating step, said resist film having a resist density and aresist surface tension; selecting a spin speed being used during thespin coating step; and selecting an ambient density of an atmospheresurrounding said resist film layer during the spin coating step.
 4. Themethod according to claim 1, wherein providing a spin coating modelfurther comprises providing a spin coating model that includes arelationship between said modeled resist film height and said spinspeed, said nominal resist film height, said resist density, said resistsurface tension, said characteristic feature size, and said ambientdensity.
 5. The method according to claim 4, wherein said spin coatingmodel further comprises a relationship between a convolution of saidtopography and a spin coating kernel function.
 6. The method accordingto claim 5, wherein said spin coating kernel function depends on theposition on said substrate and is evaluated for each of said pluralityof test points on said substrate.
 7. The method according to claim 6,wherein said spin coating kernel function further depends on said spinspeed, said nominal resist film height and said resist surface tension.8. The method according to claim 4, wherein said atmosphere is air andsaid ambient density of said atmosphere is the density of air.
 9. Themethod according to claim 8, wherein said spin coating kernel functionfurther describes the effect of radially and tangentially actingshearing stresses that is due to the air flow above the substrate. 10.The method according to claim 3, wherein the step of selecting a resistfilm being used during the spin coating step is performed by using asolvent within said resist film.
 11. The method according to claim 3,wherein said spin coating model determines said film height modulationsand a content of said solvent in an area around said plurality of testpoints, said area being smaller than the surface of said substrate. 12.The method according to claim 11, wherein said modeled resist filmheight is determined as a function of the position on the substrate foreach of said areas.
 13. The method according to claim 1, wherein saidmodeled resist film height is determined by approaching a steady state,said steady state being described by a solvent content of said resistfilm being fully evaporated.
 14. The method according to claim 13,wherein said steady state is described by a position dependent spincoating kernel function.
 15. A method for compensating film heightmodulations in spin coating of a resist film layer, the methodcomprising: providing a desired layout pattern comprising a plurality ofstructural features each having characteristic feature sizes;determining a substrate topography as a result of lithographicallystructuring a substrate with said desired layout pattern in plurality ofimage fields; providing a spin coating model, said spin coating modelbeing applicable to determine a modeled resist film height based on saidsubstrate topography during a spin coating step of a resist film on saidsubstrate as a function of respective positions on said substrate;determining a modeled resist film height for each of said respectivepositions on said substrate; determining a nominal resist film height byusing said spin coating model with an unperturbed substrate topographyhaving a flat surface; determining an average resist film height basedon said nominal resist film height; determining film height modulationsbased on a difference of said average resist film height and saidmodeled resist film height for each of said respective positions; andoptimizing said desired layout pattern by implementing furtherstructural elements in order to form an optimized mask pattern byminimizing said film height modulations.
 16. The method according toclaim 15, wherein said spin coating model determines said modeled resistfilm height for each of said respective positions as a perturbationbeing caused by said substrate topography in an area around saidrespective positions.
 17. The method according to claim 15, wherein saidspin coating model determines said modeled resist film height byinclusion of air shear.
 18. The method according to claim 15, whereinsaid spin coating model determines said modeled resist film height bycalculating a solvent content inside said area around said respectivepositions.
 19. The method according to claim 15, wherein said spincoating model further comprises a relationship between a convolution ofsaid topography and a spin coating kernel function.
 20. The methodaccording to claim 19, wherein said spin coating kernel function dependson the position on said substrate and is evaluated for each of saidplurality of test points on said substrate.
 21. The method according toclaim 20, wherein said spin coating kernel function further depends onsaid spin speed, said nominal resist film height and said resist surfacetension.
 22. The method according to claim 21, wherein said atmosphereis air and said ambient density of said atmosphere is the density ofair.
 23. The method according to claim 22, wherein said spin coatingkernel function further describes the effect of radially andtangentially acting shearing stresses that is due to the air flow abovethe substrate.
 24. The method according to claim 23, wherein said spincoating kernel function further describes a finally approached steadystate that allows predicting film height modulations after spin coatingover topographies.
 25. The method according to claim 15, wherein saidspin coating model determines said modeled resist film height usingpartial differential equations for a time evolution of the resist filmheight and the local solvent mass fraction.
 26. The method according toclaim 15, wherein said solvent content is varied so as to minimize saidmodeled resist film height.
 27. The method according to claim 25,wherein said spin coating model determines said modeled resist filmheight using partial differential equations, said equations beingformulated in terms of a velocity field beneath the resist film surface.28. The method according to claim 27, wherein said velocity field isderived on the basis of Navier-Stokes equations for an incompressibleNewtonian resist liquid.
 29. The method according to claim 28, whereinsaid velocity field further describes the influences of solventevaporation, resist film curvature and surface tension gradients on thefilm height and the local solvent content as appropriate boundaryconditions.
 30. The method according to claim 29, wherein viscosity ofsaid resist film and a surface tension coefficient of said resist filmare described by an averaged solvent fraction.
 31. A method forcompensating film height modulations in spin coating of a resist filmlayer, the method comprising the steps of: providing a desired layoutpattern comprising a plurality of structural features each havingcharacteristic feature sizes; determining a substrate topography as aresult of lithographically structuring a substrate with said desiredlayout pattern in plurality of image fields, said image fields having anactive area and a surrounding kerf area; providing a spin coating model,said spin coating model being applicable to determine a modeled resistfilm height based on said substrate topography during spin coating of aresist film; determining a nominal resist film height by using said spincoating model with an unperturbed substrate topography having a flatsurface; selecting a plurality of test points on said substrate;determining film height modulations based on a difference of saidnominal resist film height and said modeled resist film height for eachof said test points; and optimizing said desired layout pattern byimplementing further structural elements in order to form an optimizedmask pattern by minimizing said film height modulations.
 32. The methodaccording to claim 31, wherein said test points are selected within saidkerf area.
 33. The method according to claim 32, wherein said furtherstructural elements are associated to said kerf area.
 34. The methodaccording to claim 31, wherein said test points are selected within saidactive area.
 35. The method according to claim 34, wherein said furtherstructural elements are associated to said active area.
 36. The methodaccording to claim 31, wherein said further structural elements areidentical for each of said plurality of image fields.
 37. The methodaccording to claim 31, wherein said further structural elements are atleast partially different for each of said plurality of image fields.38. The method according to claim 31, further comprising: providing alithographic mask in accordance with said optimized mask pattern;structuring said substrate using said lithographic mask; and spincoating a resist film onto said substrate.
 39. The method according toclaim 38, wherein said structuring said substrate using saidlithographic mask is performed by photolithographic projection.
 40. Themethod according to claim 38, wherein said structuring said substrateusing said lithographic mask is performed by electron beam lithographie.